CONCENTRATION 329 with its long axis at right angles to the direction of fall, then since its advance causes the fluid to flow m a multitude of radial lines across the anterior face of the grain, and this sets the fluid in motion beyond the edge of the grain, the anterior pressure is largely used up in producing motion and very little is recovered in minus posterior pressure. If the pressure which the moving grain exerts is equated with the expression for the unbalanced force which has already been given and the value of 7 is transposed there is obtained for water, sp. gr. 1, the expression for V of 2.67 [D(s' —1)] ^ for a sphere. Rittingerjwho investigated the fall law experimentally gives the value of V as 2.44 [D(s' -'1)1 ^ both the diameter D and the velocity Fbeing in meters. These results are for grains of irregular shape. In the sphere formula the fall velocity is in feet and the diameter D in inches. By referring to the curve mentioned on page 328 it will be seen that there is an inflection in it. This is the point where owing to slow rate of subsidence owing to reduction in size the resistance due to the viscosity of the fluid becomes of more moment than that offered by the fluid in being' pushed aside. For small particles falling through fluids and gases the famous Stokes formula applies. Its application for air and gases has already been touched upon (p. 225). In deriving this formula the unbalanced force acting on the particle is the value which has already been derived in preceding paragraphs. The resistance of a small spherule is given by the expression GnrkV, where k is the coefficient of inner viscosity and r the radius of the spherule. The mathematical steps for obtaining this expression are very involved.1 Written in the absolute system and with the proper transformations 7 becomes 2/9fcr2(s' — s)0. This formula of course has no application in jigging as it has to do with fall velocities for particles very much below the lower range size for this operation. Kichards has found that the critical velocity or point below which the resistance tends to vary directly as the velocity rather than as the square of the velocity is to be found for galena and quartz at 0.13 mm. diameter and 63 mm. velocity per second for the former and 0.20 mm. diameter and 28 mm. velocity for the quartz. Types of Jigs.—Jigging is either done in water or air as a medium. Of the wet jigs the Harz or fixed-sieve type is the most commonly used both for ore and coal. Jigs in which the screen is moved up and down in a tank of water now have no application in the United States outside of the laboratory. Hand jigs employing this mode of operation are sometimes used for transient operations. In the Hancock jig the screen is movable and has a differential motion as to the horizontal component of the stroke as well as an up and down component. The horizontal component of the stroke moves the ore through the jig. This jig acts on the principles which have already been touched upon and on the factor of interstitial action the second principle under this section and described more fully under shaking tables at a later point. The effect of combining the two principles is to do away with the necessity of grading before jigging or the tendency within certain limits of size is to concentration according to specific gravity. The effect of mechanically advancing the material through the jig gives great capacity. At one place nine Hancock jigs displaced 132 Harz jigs occupying a floor space of 15,000 sq. ft. The nine Hancock jigs occupied 4,000 sq. ft. of space. Air Jigs.—Jigs using air as a pulsing medium work under the disadvantage that they require close sizing to effect a concentration. If two substances of specific gravity s' and s" fall through a fluid or gas with equal velocities then their diameters are in the proportion D'/D" = s" — s/sr — s where s is the specific i Consult Kirchoff's "Mathematical Physics" and Lamb's "Motion of fluids."